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Superscaling predictions for NC and CC quasi-elastic neutrino-nucleus scattering
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arXiv:1012.4265v1 [nucl-th] 20 Dec 2010
Superscaling predictions for NC and CC quasi-elastic neutrino-nucleus scattering
J.E. Amaroa, M.B. Barbarob, J.A. Caballeroc, T.W. Donnellyd,
aDepartamento de F´ısica At´omica, Molecular y Nuclear, Universidad de Granada, Granada 18071, Spain b
Dipartimento di Fisica Teorica, Universit`a di Torino and INFN, Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy
cDepartamento de F´ısica At´omica, Molecular y Nuclear,
Universidad de Sevilla, Apdo.1065, 41080 Sevilla, Spain and
dCenter for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Quasielastic double differential neutrino cross sections can be obtained in a phenomenological model based on the superscaling behavior of electron scattering data. In this talk the superscaling approach (SuSA) is reviewed and its validity is tested in a relativistic shell model. Results including meson exchange currents for the kinematics of the MiniBoone experiment are presented.
PACS numbers: 25.30.Pt,24.10.-i,25.30.Fj
Analysis of inclusive (e, e′) data have demonstrated
that at energy transfers below the quasielastic (QE) peak superscaling is fulfilled rather well [1]—[3]. The
gen-eral procedure consist on dividing the experimental (e, e′)
cross section by an appropriate single-nucleon cross sec-tion to obtain the experimental scaling funcsec-tion f (ψ), which is then plotted as a function of the scaling vari-able ψ for several kinematics and for several nuclei. If the results do not depend on the momentum transfer q, we say that scaling of the first kind occurs. If there is not dependence on the nuclear species, one has scaling of the second kind. The simultaneous occurrence of scaling of both kinds is called superscaling. The Super-Scaling approach (SuSA) is based on the assumed universality of the scaling function for electromagnetic and weak inter-actions [4].
The superscaling property is exact in the relativistic
12C fL 0.8 0.6 0.4 0.2 16 O fL 0.8 0.6 0.4 0.2 40Ca ψ fL 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 0.8 0.6 0.4 0.2 q = 0.5 GeV/c fL 0.8 0.6 0.4 0.2 q = 0.7 GeV/c fL 0.8 0.6 0.4 0.2 q = 1 GeV/c fL 0.8 0.6 0.4 0.2 q = 1.3 GeV/c fL 0.8 0.6 0.4 0.2 q = 1.5 GeV/c ψ fL 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 0.8 0.6 0.4 0.2
FIG. 1: Scaling properties of the SR shell model. Left: scal-ing of the first kind. Curves for q = 0.5, 0.7, 1, 1.3, 1.5 GeV collapse into one. Right: scaling of the second kind. Curves for12
C,16
O and40
Ca collapse into one.
Fermi gas model (RFG) by construction, and it has been
tested in more realistic models of the (e, e′) reaction [5]–
[7]. A study of superscaling in the semirelativistic (SR) continuum shell model with Woods-Saxon (WS) mean potential is summarized in Fig. 1. There we show the
longitudinal scaling function defined as fL = RL/GL,
where RL is the longitudinal response function and GL
is a single-nucleon factor. When fL(ψ) is plotted for
various values of the momentum transfer and for several closed-shell nuclei, all the curves approximately collapse into one. Small violations of scaling are seen at low values of ψ, coming from the low-energy potential resonances for q = 0.5 GeV/c, which disappear for higher q values.
While the SR shell model superscales, it does not re-produce the experimental data of the phenomenological scaling function extracted from the longitudinal QE elec-tron scattering response. The WS potential used to de-scribe the final-state interaction (FSI) of the ejected pro-ton does not incorporate the appropriate reactions mech-anisms. A further improvement of the FSI consist in us-ing the Dirac-Equation based potential plus Darwin term (DEB+D). The DEB potential is obtained from the Dirac
equation for the upper component ψup(r) = K(r, E)φ(r),
where the Darwin term K(r, E) is chosen in such a way that the function φ(r) satisfies a Schr¨odinger-like
equa-tion. The electromagnetic L and T scaling functions
within this model are shown in Fig. 2. We use the same relativistic Hartree potential as in the relativistic mean
field model of [5], and the scaling variable ψ′ includes
a q-dependent energy shift Es(q). Although the scaling
is not perfect, it is remarkable that our results give es-sentially the same scaling function for a wide range of q values, reproducing well the phenomenological data.
In Fig. 2 (right) we compare DEB+D and
Woods-Saxon predictions for the 12C(ν
µ, µ−) differential QE
cross section for 1 GeV incident neutrinos and for sev-eral lepton scattering angles. The SuSA reconstructed cross section is also shown, using the theoretical scaling
function extracted from (e, e′) results with the DEB+D
model (curves in Fig. 2, left). For angles above 15o the
2 ψ′ fL 3 2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 ψ′ fT 3 2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 ψ′ fL 3 2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 ψ′ fT 3 2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 q [MeV/c] Es [M eV ] 1600 1400 1200 1000 800 600 400 80 70 60 50 40 30 20 θ = 15o ǫ = 1 GeV 1000 950 900 850 800 60 50 40 30 20 10 0 θ = 30o ǫ = 1 GeV 1000 900 800 700 600 18 16 14 12 10 8 6 4 2 0 θ = 45o 1000 900 800 700 600 500 400 7 6 5 4 3 2 1 0 θ = 60o 900 800 700 600 500 400 300 3 2.5 2 1.5 1 0.5 0 θ = 75o d σ d Ω ′dǫ ′ [1 0 − 1 5fm 2/M eV ] 800 700 600 500 400 300 200 1.6 1.2 0.8 0.4 0 θ = 90o 800 700 600 500 400 300 200 1 0.8 0.6 0.4 0.2 0 θ = 105o 700 600 500 400 300 200 100 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 θ = 120o 600 500 400 300 200 100 0.6 0.5 0.4 0.3 0.2 0.1 0 θ = 135o ǫ′[MeV] 600 500 400 300 200 100 0.4 0.3 0.2 0.1 0 θ = 150o ǫ′[MeV] 500 400 300 200 100 0.3 0.2 0.1 0
FIG. 2: Left: Scaling of 1st kind with DEB+D potential com-pared with experimental data. q = 0.5, 0.7 1.0, 1.3 and 1.5 GEV/c. Right: Test of SuSA in the SR shell model for the
12
C(νµ, µ−) reaction with neutrino energy ǫ = 1 GeV. Dotted: Woods-Saxon potential. Solid: DEB+D potential. Dashed: SuSA reconstruction from the computed (e, e′) scaling
func-tion. ;N=20 0 800 700 600 500 400 0.8 0.6 0.4 0.2 0 ;N=60 0 400 350 300 250 200 150 100 50 0 0.8 0.6 0.4 0.2 0 ;N=20 0 TN[MeV℄ d 2 =d =dE N [10 15 f m 2 =sr = M eV ℄ 800 700 600 500 400 0.08 0.06 0.04 0.02 0 ;N=60 0 TN[MeV℄ 400 350 300 250 200 150 100 50 0 0.4 0.3 0.2 0.1 0 Strange g s A = 0:2 s=0:55 Nostrange ;p=20 0 800 700 600 500 400 0.8 0.6 0.4 0.2 0 ;p=60 0 400 300 200 100 0 0.8 0.6 0.4 0.2 0 ;p=20 0 Tp[MeV℄ d 2 =d p =dE p [10 15 f m 2 =sr = M eV ℄ 800 700 600 500 400 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 ;p=60 0 Tp[MeV℄ 400 300 200 100 0 0.5 0.4 0.3 0.2 0.1 0
FIG. 3: Neutral current proton knock-out cross section from
12
C. Left: RFG (solid lines), RFG with factorization (dotted lines), and SuSA (dashed lines). Right: study of nucleon strangeness effect. H1 H2 P′ 1 P′2 Q K2 (a) H1 P′ 1 H2 P′ 2 Q K1 (b) H1 P′ 1 H2 P′ 2 Q K1K2 (c) H1 H2 P′ 1 P2′ Q K2 (d) H1 P′ 1 H2 P′ 2 Q K1 (e) H1 H2 P′ 1 P2′ Q K2 (f) H1 P′ 1 H2 P′ 2 Q K1 (g)
FIG. 4: Diagrams contributing to the MEC
quite reasonable, with small error.
The superscaling approach has been extended to in-clusive neutrino scattering via the weak neutral current [8]. In the (ν, N ) reaction a nucleon is detected and the final neutrino must be integrated. We approximate the u-channel inclusive scattering cross section
dσ
dΩNdpN
≃σ(u)snF
(u)(ψ(u)). (1)
3 0.8 < cos θ < 0.9 Tµ(GeV) d 2σ /d co s θ/ d Tµ (1 0 − 3 9cm 2/G eV ) 2 1.5 1 0.5 0 25 20 15 10 5 0 data SuSA+MEC 0.8 < cos θ < 0.9 Tµ(GeV) d 2σ /d co s θ/ d T µ (1 0 − 3 9cm 2/G eV ) 2 1.5 1 0.5 0 25 20 15 10 5 0 0.7 < cos θ < 0.8 Tµ(GeV) d 2σ /d co s θd Tµ (1 0 − 3 9cm 2/G eV ) 2 1.5 1 0.5 0 25 20 15 10 5 0 0.7 < cos θ < 0.8 Tµ(GeV) d 2σ /d co s θd T µ (1 0 − 3 9cm 2/G eV ) 2 1.5 1 0.5 0 25 20 15 10 5 0 0.6 < cos θ < 0.7 Tµ(GeV) d 2σ /d co s θ/ d T µ (1 0 − 3 9cm 2/G eV ) 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 20 15 10 5 0 0.6 < cos θ < 0.7 Tµ(GeV) d 2σ /d co s θ/ d T µ (1 0 − 3 9cm 2/G eV ) 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 20 15 10 5 0 0.5 < cos θ < 0.6 Tµ(GeV) d 2σ /d co sθ /d T µ (1 0 − 3 9cm 2/G eV ) 1.2 1 0.8 0.6 0.4 0.2 0 18 16 14 12 10 8 6 4 2 0 0.5 < cos θ < 0.6 Tµ(GeV) d 2σ /d co sθ /d T µ (1 0 − 3 9cm 2/G eV ) 1.2 1 0.8 0.6 0.4 0.2 0 18 16 14 12 10 8 6 4 2 0 0.4 < cos θ < 0.5 Tµ(GeV) d 2σ /d co s θ/ d Tµ (1 0 − 3 9cm 2/G eV ) 1 0.8 0.6 0.4 0.2 0 18 16 14 12 10 8 6 4 2 0 0.4 < cos θ < 0.5 Tµ(GeV) d 2σ /d co s θ/ d T µ (1 0 − 3 9cm 2/G eV ) 1 0.8 0.6 0.4 0.2 0 18 16 14 12 10 8 6 4 2 0 0.3 < cos θ < 0.4 Tµ(GeV) d 2σ /d co s θd Tµ (1 0 − 3 9cm 2/G eV ) 1 0.8 0.6 0.4 0.2 0 14 12 10 8 6 4 2 0 0.3 < cos θ < 0.4 Tµ(GeV) d 2σ /d co s θd T µ (1 0 − 3 9cm 2/G eV ) 1 0.8 0.6 0.4 0.2 0 14 12 10 8 6 4 2 0
FIG. 5: Computed QE (νµ, µ−) cross section compared to Miniboone data. Left: SuSA. Right: SuSA + MEC
approximation in the RFG, and therefore one could ex-tend the scaling analysis used for CC reactions to NC scattering. In Fig. 3 we show examples of SuSA pre-dictions of NC cross sections using the phenomenological (e, e′) scaling function.
Recently the effect of meson exchange currents (MEC)
in (e, e′) for high momentum transfer has been
investi-gated [9, 10], and the two-particle emission (2p-2h) di-agrams of Fig. 4 have been extended to the CC weak interaction sector [11], in order to explore the role of MEC in neutrino reactions. In Fig. 5 we show that inclusion of 2p-2h contributions yields results for the QE
(νµ, µ−) cross section that are comparable with the
re-cent MiniBooNE collaboration data [12] for θ ≤ 50o, but
lie below the data at larger angles where the predicted cross sections are smaller. The inclusion of the correla-tion diagrams which are required by gauge invariance [10] plus other relativistic effects might improve the agree-ment with the data.
This work was partially supported by DGI (Spain): FIS2008-01143, FIS2008-04189, by the Junta de An-daluc´ıa, and part (TWD) by U.S. Department of Energy under cooperative agreement DE-FC02-94ER40818.
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